Symmetric and Skew Symmetric Matrices

Symmetric Matrix:

A square matrix A is symmetric if it is equal to its transpose: A=AT

  • Properties:
    • The elements along the main diagonal remain unchanged when transposed.
    • aij=aji for all i and j.
    • It is always a square matrix (number of rows = number of columns).

Example of Symmetric Matrix:

A=[257539791]

In this case, A=AT, which makes it a symmetric matrix.

Skew-Symmetric Matrix:

A square matrix A is skew-symmetric if it is equal to the negative of its transpose: A=AT

  • Properties:
    • The elements along the main diagonal are zero.
    • aij=aji for all i and j.
    • It is always a square matrix (number of rows = number of columns).

Example of Skew-Symmetric Matrix:

A=[073702320]

In this case, A=AT, which makes it a skew-symmetric matrix.

Properties of Symmetric and Skew-Symmetric Matrices:

  1. Sum of Symmetric Matrices: The sum of two symmetric matrices is also symmetric. A+B=Cwhere A, B, and C are symmetric matrices.

  2. Sum of Skew-Symmetric Matrices: The sum of two skew-symmetric matrices is also skew-symmetric. A+B=C where A, B, and C are skew-symmetric matrices.

  3. Product with Scalar: Both symmetric and skew-symmetric matrices preserve their property when multiplied by a scalar.